In this post I want to describe a new gauge for the ordinarity of abelian varieties over p-adic fields. The definition is rather simple; the part which actually requires work is showing it cuts out a well-behaved locus in the relevant Siegel moduli spaces. This locus is most naturally defined at infinite level, where it carries some remarkable extra structures (most notably, the functions defined below). These observations lead naturally to an alternate construction of sheaves of overconvergent modular forms which absorbs any explicit use of difficult (i.e. torsion) p-adic Hodge theory into Scholze’s construction of his infinite-level Shimura varieties. This approach is the nonarchimedean analogue of the classical “” approach to holomorphic modular forms.

Let be a complete algebraically closed extension of , with ring of integers . Given any smooth proper rigid space , we have a general Hodge-Tate spectral sequence

Taking an abelian variety and recalling the identification , the edge map gives (after twisting and replacing by ) a natural Hodge-Tate map

Let denote the -span of the image of the Hodge-Tate map. For any rational and any , we write for the induced map . For later use, we note that the invariant differentials on the Néron model of define a canonical -lattice , and work of Faltings and Fargues implies the inclusions .

Definition.

i. A -dimensional abelian variety is w-ordinary if , where .

ii. Given a rigid space and a family of abelian varieties , is -ordinary if is w-ordinary at all geometric points .

Note that is classically ordinary if and only if it is -ordinary.

Given a -ordinary abelian variety and any , the kernel of the map is an étale group scheme isomorphic to , which we term the pseudocanonical subgroup. We say a trivialization is strict if the induced map trivializes the pseudocanonical subgroup.

Let be the symplectic group for the usual matrix , with a typical element. Let be the involution of given by . Let be the Siegel parabolic defined by , its unipotent radical, embedded by , so . Inside we have the usual open subgroups . Let be the principal congruence subgroup of level , and let be any open subgroup contained in for some . Let denote the Siegel modular variety over parametrizing principally polarized abelian varieties with -level structure. We shall consider levels of the form for some fixed open ; we write for and for . Let denote the minimal compactification, and write , , , etc. for the associated rigid analytic (equivalently, locally tft adic) spaces over , so the -points of are in bijection with isomorphism classes of quadruples , where is a -dimensional abelian variety, is a principal polarization, is a -level structure, and is a symplectic trivialization. In what follows I’ll suppress and , conflating a point with the associated pair . We have an action of on given by where

Let denote the perfectoid space constructed by Scholze in Chapter III of his torsion paper, so we have a -equivariant identification

and a -equivariant Hodge-Tate period map

whose definition we recall below.

Theorem A.

i. There is a canonical open rigid subspace parametrizing principally polarized -ordinary abelian varieties with -level structure, and this inclusion factors through the composite of a natural open immersion with the natural morphism .

ii. For any open , the space parametrizes principally polarized -ordinary abelian varieties equipped with a -level structure and a strict -level structure. The spaces admit canonical affinoid minimal compactifications , and for the natural morphism (resp. ) is finite etale (resp. finite).

iii.In the Shimura variety of infinite level we have -stable open subspaces and such that

and

The space is affinoid perfectoid, and there is a natural -stable affinoid subdomain such that .

I’ve stated the results in this theorem in the opposite order from their proofs: the simple description of in terms of the Hodge-Tate period map is the key ingredient in our comprehension of these -ordinary loci. To explain this, let with basis and the usual symplectic form , and let be the flag variety with parametrizing maximal isotropic flags . Given a point , the Hodge-Tate map for sits in a short exact sequence

and Scholze’s Hodge-Tate period map sends to the flag

Inside we have the (Zariski-open and dense) big cell where for some (any) basis of the matrix is invertible. Over we have canonical global sections () characterized by the fact that for any point , the vectors form the unique basis of with (the canonical basis of ). The matrix is symmetric. For rational, let be the subspace of cut out by the condition

This is naturally an affinoid, with , and it’s not hard to check that is -stable – in fact, we have the explicit formula

for .

Definition. We set . The fundamental periods are the global sections of the structure sheaf of defined by .

The matrix is the nonarchimedean version of the usual coordinates on Siegel upper half-space. The functions aren’t exactly coordinates anymore (since isn’t injective), but they still carry an enormous amount of information.

The theorem above follows from the following two propositions.

Proposition B. Given a point with associated abelian variety and -subgroup , then is -ordinary and is pseudocanonical if and only if some (any) point lifting is contained in .

Proof. Let be the trivialization associated with , and set . If , then for with , and after replacing by a -translate we may arrange so that . Now each has trivial image in for , so the image of each in for (with ) lies in the kernel of , and therefore is -ordinary. By definition the images of these ‘s in trivialize the -subgroup , but the former clearly generate the pseudocanonical subgroup.

The converse direction is similar.

Proposition C. The space is affinoid perfectoid, and comes by pullback from an affinoid .

Proof. We briefly recall the Plucker cells of . For a subset of cardinality , let denote the wedge of the ‘s for (in the natural ordering). Given with associated , then choosing any basis of and writing

defines a point of if and only if for all . (Alternately, each is a well-defined meromorphic function on , and is cut out by the condition .)

Now setting and letting be the canonical basis of the flag associated with a point , the following things are clear:

–,

– where ,

-each is a homogeneous polynomial in the ‘s of degree .

In particular, and each extends to a section of . Now the following are some of the main results in Scholze’s torsion paper:

i. the subspaces are affinoid perfectoid,

ii. we may find some such that is the preimage of an affinoid for all ,

iii. writing and defining as the -adic completion of , then .

Taking , we see that each canonically extends to a section of , and the condition

cuts out as a rational subdomain of . One of the important basic results on perfectoid spaces is that rational subdomains of affinoid perfectoids are themselves affinoid perfectoid, so this proves the first clause of the proposition. Now for some sufficiently large depending on , we may choose some large and some such that , and then the condition

again cuts out (cf. Remark 2.8 in “Perfectoid spaces”) and visibly descends to a condition cutting out a rational subdomain whose preimage at infinite level is . Since is -stable, is as well.

It remains to descend down to level . Given a finite group acting on a rigid space , we say a finite morphism is a –covering if is -equivariant for the trivial action on , is generically of degree , and acts transitively on the fibers of (Is there a simpler definition? I don’t want to assume that is etale.)

Lemma. If is a -covering with normal, there is a functorial bijection between affinoid subdomains and -stable affinoid subdomains realized explicitly by the functors

and

Proof. By the discussion in 6.3.3 of BGR, is affinoid and is a module-finite and integral -algebra. By Corollary 9.4.4/2 in BGR,

is affinoid and module-finite over . To see that these functors are mutually inverse, note that is finite and birational, and therefore an isomorphism by the normality of . The identification is an easy consequence of the set-theoretic surjectivity of the natural map .

Applying this lemma with , , , the obvious map, and , we obtain as . (The finiteness of and the normality of are basic properties of the minimal compactifications.) Now the rest is easy: we define as the preimage of , and then . The twiddly relation in part iii. of the theorem follows immediately from a result in Scholze-Weinstein.

It’s natural to wonder how -ordinarity compares with the usual measure of ordinarity, namely the size of the Hasse invariant. For , let denote the locus where , so is the ordinary locus. Since and both give a sequence of shrinking affinoid neighborhoods of , they must intertwine “for compacity reasons”. Here’s a more concrete statement.

Theorem D. If and for some integer , then is -ordinary for

In particular, for we have for the same .

This crucially uses results of Fargues on the canonical subgroup (plus some fiddling to deal with the boundary). There’s a similar statement for .

Let me end with a teaser, which I’ll discuss more carefully in a followup post. Set , so we’re in the realm of modular curves (and I’ll replace by as is customary). Let be a finite extension of , and choose a continuous character . Let be the natural map, so for any affinoid the preimage is a perfectoid space, and I’ll write for the global sections of its structure sheaf. (Note: might not be affinoid perfectoid! But if is a rational subdomain then it will be, so by Gerritzen-Grauert has a finite covering by affinoid perfectoids.) There’s a left action of on , inducing a right action on . Now there’s a unique fundamental period , and there’s a rational depending on such that for any , the expression gives a well-defined element of for any , .

Definition. We define a sheaf on by the rule

for any rational subdomain .

Theorem E. The sheaf is a line bundle on , and coincides with the sheaf of overconvergent forms of weight defined in Pilloni’s Annales de l’Institut Fourier paper.